用数学归纳法证明，对于所有正整数 $n$，$5^{3n}+32^n-33$ 都能被 $31$ 整除。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

先验n=1，再比较相邻表达式，差的两个系数都是31的倍数。

Let $F_n=5^{3n}+32^n-33$. For $n=1$,

so the statement holds.

Assume that $F_k$ is divisible by $31$ for some positive integer $k$. Then

This difference is divisible by $31$. Since $F_k$ is divisible by $31$, so is $F_{k+1}$.

The statement is true for $n=1$ and truth for $n=k$ implies truth for $n=k+1$. Thus, by mathematical induction, $5^{3n}+32^n-33$ is divisible by $31$ for every positive integer $n$.